Electronic Communications in Probability

The glassy phase of the complex branching Brownian motion energy model

Lisa Hartung and Anton Klimovsky

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We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase'' – the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas (2013), where the uncorrelated case was treated. Inparticular, our result covers the case of the real-valued BBM energy model at complex temperatures.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 78, 15 pp.

Accepted: 27 October 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Gaussian processes branching Brownian motion logarithmic correlations random energy model phase diagram extremal processes cluster processes multiplicative chaos

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Hartung, Lisa; Klimovsky, Anton. The glassy phase of the complex branching Brownian motion energy model. Electron. Commun. Probab. 20 (2015), paper no. 78, 15 pp. doi:10.1214/ECP.v20-4360. https://projecteuclid.org/euclid.ecp/1465321005

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